Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors

A Fourier modal method for analysing crossed anisotropic gratings is presented without any restrictions on the permittivity and permeability tensors of the medium in the grating region and the angle between the two periodic directions. A skew Cartesian coordinate system with skew angles both in and out of the grating plane is used in the mathematical derivation, which gives the formulation a greater generality. However, the facets of the crossed gratings are required to be parallel to the skew coordinate surfaces. Correct Fourier factorization of Maxwell's equations is carried out with the aid of two efficient operators that greatly simplify the otherwise complicated notation. Numerical curves are presented to demonstrate the convergence of the method, and numerical values are tabulated to provide reference data.